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Fig. 4. Entropy s“V, a), and enthalpy ”4V, b), of humid air in equi- librium with seawater, Eq. (6.18), at sea-level pressure as a func- tion of the sea-surface temperature (SST) for salinities 0, 35 and 120 g/kg. The related humidity is shown in Fig. 3. Entropy and enthalpy of dry air are plotted for comparison. The last term of Eq. (6.22) is the latent-heat part of the heat capacity caused by the transfer of water between the liquid and the gas phase when the temperature is changing at con- stant pressure.

Figure 4 Entropy s“V, a), and enthalpy ”4V, b), of humid air in equi- librium with seawater, Eq. (6.18), at sea-level pressure as a func- tion of the sea-surface temperature (SST) for salinities 0, 35 and 120 g/kg. The related humidity is shown in Fig. 3. Entropy and enthalpy of dry air are plotted for comparison. The last term of Eq. (6.22) is the latent-heat part of the heat capacity caused by the transfer of water between the liquid and the gas phase when the temperature is changing at con- stant pressure.

Related Figures (55)

Fig. 1. Non-ideal behaviour (Z4V — 1), Eq. (5.3), of dry (upper panel) and saturated (lower panel) air for several values of T and P. such as con densers of desalination plants or headspaces over subglacial lakes. The most relevant restriction for the ac- curacy of Eq. (5.2) is the truncation error of the virial ex- igibly sma plies limita fraction. T the appendi of Eq. (5.2) pansion, i.e., its omitted subsequent terms proportional to A3(1— A) p3, A2(1—A)* po? and A(1— A)? 0? must be neg- l. Hyland and Wexler (1983b) require pressures ess than 5 MPa for saturated humid air, this condition im- tions to the air density in dependence of the air he comparison with experimental data shown in x, Fig. D1, suggests that the current formulation can safely be used up to densities of 100 kgm~? higher cros 200kgm-? of saturated humid air. Results remain reasonable beyond this point, but the uncertain third and the missing fourth and s-Virial coefficients are clearly relevant beyond Describing the deviation of the equation of state from ideal- gas behaviour, the compressibility factor Zay of humid air is defined as (van Wylen and Sonntag, 1965),
Fig. 1. Non-ideal behaviour (Z4V — 1), Eq. (5.3), of dry (upper panel) and saturated (lower panel) air for several values of T and P. such as con densers of desalination plants or headspaces over subglacial lakes. The most relevant restriction for the ac- curacy of Eq. (5.2) is the truncation error of the virial ex- igibly sma plies limita fraction. T the appendi of Eq. (5.2) pansion, i.e., its omitted subsequent terms proportional to A3(1— A) p3, A2(1—A)* po? and A(1— A)? 0? must be neg- l. Hyland and Wexler (1983b) require pressures ess than 5 MPa for saturated humid air, this condition im- tions to the air density in dependence of the air he comparison with experimental data shown in x, Fig. D1, suggests that the current formulation can safely be used up to densities of 100 kgm~? higher cros 200kgm-? of saturated humid air. Results remain reasonable beyond this point, but the uncertain third and the missing fourth and s-Virial coefficients are clearly relevant beyond Describing the deviation of the equation of state from ideal- gas behaviour, the compressibility factor Zay of humid air is defined as (van Wylen and Sonntag, 1965),
The specific entropy of humid air follows from is known in meteorology as the virtual temperature of hu- mid air (Rogers and Yau, 1989; Jacobson, 2005; Hantel and Mayer, 2006). The Gibbs function of humid air, hv, follows from its Helmholtz function (5.2) via the Legendre transform
The specific entropy of humid air follows from is known in meteorology as the virtual temperature of hu- mid air (Rogers and Yau, 1989; Jacobson, 2005; Hantel and Mayer, 2006). The Gibbs function of humid air, hv, follows from its Helmholtz function (5.2) via the Legendre transform
Fig. 2. Adiabatic lapse rate (5.26) of saturated humid air at differ- ent temperatures and pressures as indicated by the curves. Note that this lapse rate is termed “dry” in some literature. For convenience, the rate is shown per 100m altitude difference as common in me- teorology, rather than per pressure difference. For comparison, the lapse rate of wet air is shown in Fig. 7.
Fig. 2. Adiabatic lapse rate (5.26) of saturated humid air at differ- ent temperatures and pressures as indicated by the curves. Note that this lapse rate is termed “dry” in some literature. For convenience, the rate is shown per 100m altitude difference as common in me- teorology, rather than per pressure difference. For comparison, the lapse rate of wet air is shown in Fig. 7.
Fig. 3. Specific (a) and relative (b) humidity of sea air, ie., humid air in equilibrium with seawater, at sea level pressure, P=101 325 Pa, as a function of temperature for different absolute salinities as indicated by the curves. The equilibrium air fraction, A= And at the condensation point is computed from Eq. (6.4), the resulting specific humidity is g=(1— A). The relative humidity RHwmo is determined from A by the WMO definition (10.4) with the saturated air fraction AS* from Eq. (7.5). AS@ is displayed as the zero-salinity curve “0 g/kg” in (a).
Fig. 3. Specific (a) and relative (b) humidity of sea air, ie., humid air in equilibrium with seawater, at sea level pressure, P=101 325 Pa, as a function of temperature for different absolute salinities as indicated by the curves. The equilibrium air fraction, A= And at the condensation point is computed from Eq. (6.4), the resulting specific humidity is g=(1— A). The relative humidity RHwmo is determined from A by the WMO definition (10.4) with the saturated air fraction AS* from Eq. (7.5). AS@ is displayed as the zero-salinity curve “0 g/kg” in (a).
which tells us how changes in one state variable must be compensated by changes in some other state variable(s) in order to maintain thermodynamic equilibrium. corresponding to the transitivity of the binary relation “equi- librium” between phases. Using this operator, the equilib- rium condition (6.4) takes the simple form A,as[g]=0. Its total differential gives the Clausius-Clapeyron differential equation of this phase transition,
which tells us how changes in one state variable must be compensated by changes in some other state variable(s) in order to maintain thermodynamic equilibrium. corresponding to the transitivity of the binary relation “equi- librium” between phases. Using this operator, the equilib- rium condition (6.4) takes the simple form A,as[g]=0. Its total differential gives the Clausius-Clapeyron differential equation of this phase transition,
SA The specific isobaric heat capacity of sea air, c?*, is com- puted from Eq. (6.18), as Using Eqs. (6.5), (6.13) and (6.15), this expression can be rearranged to give Taking the derivatives in Eqs. (6.18) and (6.20), the prop- erty of the composite system is just the sum of the fractions of the two phases. Any additional terms cancel because of Eq. (6.4), Aas[g]=0. This is no longer the case if the sec- ond derivatives of the Gibbs function (6.17) are involved, as for the heat capacity, compressibility or thermal expansion.
SA The specific isobaric heat capacity of sea air, c?*, is com- puted from Eq. (6.18), as Using Eqs. (6.5), (6.13) and (6.15), this expression can be rearranged to give Taking the derivatives in Eqs. (6.18) and (6.20), the prop- erty of the composite system is just the sum of the fractions of the two phases. Any additional terms cancel because of Eq. (6.4), Aas[g]=0. This is no longer the case if the sec- ond derivatives of the Gibbs function (6.17) are involved, as for the heat capacity, compressibility or thermal expansion.
Fig. 5. Isobaric evaporation enthalpy of water, indicated by “O g/kg”, and of seawater with salinity 120 gkg~/ at 101 325 Pa and 40gkg—! at 5MPa and 10MPa, computed from Eq. (6.28). The salinity dependence is very weak in either case. At high pressure the validity of the current Gibbs function of seawater (3.1) is re- stricted to maxima of only 40 gkg~! and 40°C, but at atmospheric pressure it is valid up to 120 gkg~! and 80°C (IAPWS, 2008; Feis- tel, 2008). The lower temperature bounds are the particular freezing points of water or seawater.
Fig. 5. Isobaric evaporation enthalpy of water, indicated by “O g/kg”, and of seawater with salinity 120 gkg~/ at 101 325 Pa and 40gkg—! at 5MPa and 10MPa, computed from Eq. (6.28). The salinity dependence is very weak in either case. At high pressure the validity of the current Gibbs function of seawater (3.1) is re- stricted to maxima of only 40 gkg~! and 40°C, but at atmospheric pressure it is valid up to 120 gkg~! and 80°C (IAPWS, 2008; Feis- tel, 2008). The lower temperature bounds are the particular freezing points of water or seawater.
Fig. 6. Entropy of wet air, Eq. (7.10), as a function of temperature at 1013.25 hPa (a) and 500 hPa (b) between the freezing point and complete evaporation of the liquid component for dry-air fractions w' of wet air, Eq. (7.3), between 10% and 90% as indicated by the curves. The related liquid mass fractions w™ of wet air, Eq. (7.3), are shown in Fig. 9.
Fig. 6. Entropy of wet air, Eq. (7.10), as a function of temperature at 1013.25 hPa (a) and 500 hPa (b) between the freezing point and complete evaporation of the liquid component for dry-air fractions w' of wet air, Eq. (7.3), between 10% and 90% as indicated by the curves. The related liquid mass fractions w™ of wet air, Eq. (7.3), are shown in Fig. 9.
Fig. 7. Adiabatic lapse rate (7.19) of wet air at the dewpoint, i.e. containing a negligible amount of liquid water, at different temper- atures and pressures as indicated by the curves. Note that this lapse rate is termed “moist” in some literature. For convenience, the rate is shown per 100 m altitude difference as common in meteorology, rather than per pressure difference. For comparison, the lapse rate of humid air is shown in Fig. 2.
Fig. 7. Adiabatic lapse rate (7.19) of wet air at the dewpoint, i.e. containing a negligible amount of liquid water, at different temper- atures and pressures as indicated by the curves. Note that this lapse rate is termed “moist” in some literature. For convenience, the rate is shown per 100 m altitude difference as common in meteorology, rather than per pressure difference. For comparison, the lapse rate of humid air is shown in Fig. 2.
Fig. 8. The saturation curves AR, (T,P) computed from solv- ing Eq. (7.5) at the pressures 1013.25, 500, 200 and 100 hPa, as indicated, between the freezing and the boiling temperature of pure water separate the physically sensible dry-air fractions, A > AS (7, P), of the single-phase state of humid air from those of the two-phase state of wet air, wi < ast (T, P). Ona particular saturation curve the two systems coincide, Eqs. (7.23), (7.24). The liquid fraction of wet airw™ = 1—w/at (7, P), Eq. (7.3), is shown in Fig. 9.
Fig. 8. The saturation curves AR, (T,P) computed from solv- ing Eq. (7.5) at the pressures 1013.25, 500, 200 and 100 hPa, as indicated, between the freezing and the boiling temperature of pure water separate the physically sensible dry-air fractions, A > AS (7, P), of the single-phase state of humid air from those of the two-phase state of wet air, wi < ast (T, P). Ona particular saturation curve the two systems coincide, Eqs. (7.23), (7.24). The liquid fraction of wet airw™ = 1—w/at (7, P), Eq. (7.3), is shown in Fig. 9.
Fig. 9. Liquid mass fraction w of wet air, the “burden” of con- densed water, as a function of temperature at 1013.25 hPa (a) and 500 hPa (b) between the freezing point and the point at which com- plete evaporation of the liquid component occurs for dry-air frac- tions w4 of wet air, Eq. (7.3), between 10% and 90% as indicated by the curves. The complement (1 — wW) is the gaseous humid-air fraction of wet air.
Fig. 9. Liquid mass fraction w of wet air, the “burden” of con- densed water, as a function of temperature at 1013.25 hPa (a) and 500 hPa (b) between the freezing point and the point at which com- plete evaporation of the liquid component occurs for dry-air frac- tions w4 of wet air, Eq. (7.3), between 10% and 90% as indicated by the curves. The complement (1 — wW) is the gaseous humid-air fraction of wet air.
Fig. 10. Specific (a) and relative (b) humidity of sea air, ie., humid air in equilibrium with seawater, at sea-level pressure, P=101 325 Pa, as a function of temperature for different absolute salinities as indicated by the curves. The equilibrium air fraction A is computed from Eq. (6.4); the resulting specific humidity is q=(1—A). The relative humidity RHwmo is determined from A by the WMO definition (10.4) with the saturated air fraction AS*' given by Eq. (8.7). The specific humidity related to AS* is displayed as the bold curve “Ice” a), indicating the saturation of sea-ice air due to ice formation below the freezing temperature at the particular salinity.
Fig. 10. Specific (a) and relative (b) humidity of sea air, ie., humid air in equilibrium with seawater, at sea-level pressure, P=101 325 Pa, as a function of temperature for different absolute salinities as indicated by the curves. The equilibrium air fraction A is computed from Eq. (6.4); the resulting specific humidity is q=(1—A). The relative humidity RHwmo is determined from A by the WMO definition (10.4) with the saturated air fraction AS*' given by Eq. (8.7). The specific humidity related to AS* is displayed as the bold curve “Ice” a), indicating the saturation of sea-ice air due to ice formation below the freezing temperature at the particular salinity.
Fig. 11. Equilibrium pressure P, panel (a), and temperature 7, panel (b), as a function of the mass fraction A of dry air in humid air of a parcel containing liquid water, ice and humid air. At A=0, the curve ends in the common triple point of water at about 611 Pa and 273.16K. Note that the dissolution of air in liquid water is not considered here which may have comparable additional effects on the freezing temperature (Doherty and Kester, 1974; IAPWS, 2004; Feistel, 2008a).
Fig. 11. Equilibrium pressure P, panel (a), and temperature 7, panel (b), as a function of the mass fraction A of dry air in humid air of a parcel containing liquid water, ice and humid air. At A=0, the curve ends in the common triple point of water at about 611 Pa and 273.16K. Note that the dissolution of air in liquid water is not considered here which may have comparable additional effects on the freezing temperature (Doherty and Kester, 1974; IAPWS, 2004; Feistel, 2008a).
Fig. 12. Entropy of ice air as a function of temperature at 1013.25 hPa (a) and 500 hPa (b) between —100°C and the melt- ing point of the solid component for dry-air fractions w“ of ice air between 10% and 90% as indicated by the curves.
Fig. 12. Entropy of ice air as a function of temperature at 1013.25 hPa (a) and 500 hPa (b) between —100°C and the melt- ing point of the solid component for dry-air fractions w“ of ice air between 10% and 90% as indicated by the curves.
Fig. 13. Saturation curves ASL (T, P) computed by solving Eq. (9.2) at the pressures 1013.25, 500, 200 and 100 hPa, as in- dicated. Between —60°C and the melting temperature of ice, these curves separate the physically reasonable dry-air fractions, A> ASL(T, P), of the single- phase state of humid air from those of the two-phase state of ice air, w“ < ar, P). At the saturation point, the two systems coincide.
Fig. 13. Saturation curves ASL (T, P) computed by solving Eq. (9.2) at the pressures 1013.25, 500, 200 and 100 hPa, as in- dicated. Between —60°C and the melting temperature of ice, these curves separate the physically reasonable dry-air fractions, A> ASL(T, P), of the single- phase state of humid air from those of the two-phase state of ice air, w“ < ar, P). At the saturation point, the two systems coincide.
Fig. 14. Mass fraction A of dry air in humid air as a function of temperature at nomal pressure, 101 325 Pa, for different relative hu- midities (WMO definition, Eq. 10.4) as indicated by the curves from 10% to 100% (dewpoint). Specific humidity is g=1— A. Below the dewpoint curve (100%), liquid water is present, i.e., the stable state is wet air rather than humid air.
Fig. 14. Mass fraction A of dry air in humid air as a function of temperature at nomal pressure, 101 325 Pa, for different relative hu- midities (WMO definition, Eq. 10.4) as indicated by the curves from 10% to 100% (dewpoint). Specific humidity is g=1— A. Below the dewpoint curve (100%), liquid water is present, i.e., the stable state is wet air rather than humid air.
Fig. 15. Vertical isentropic sea-air profiles of density overlying seawater at standard-ocean salinity (35.16504 g kg~!) and sea-level pressure (101 325 Pa) for different SST values between 0 and 40 °C as indicated. The related isentropic condensation levels (ICL) of liquid water and ice (bold lines) are also shown.
Fig. 15. Vertical isentropic sea-air profiles of density overlying seawater at standard-ocean salinity (35.16504 g kg~!) and sea-level pressure (101 325 Pa) for different SST values between 0 and 40 °C as indicated. The related isentropic condensation levels (ICL) of liquid water and ice (bold lines) are also shown.
Fig. 16. Vertical isentropic sea-air profiles (almost vertical thin lines) of temperature for different SST values between 0 and 40°C at standard-ocean salinity and sea-level pressure of 101 325 Pa, to- gether with the related isentropic condensation levels (ICL) of liq- uid water and ice (bold lines). Below the freezing temperature of seawater, sea air is saturated at the sea surface. The slope of the isentropes is given by the adiabatic lapse rate, Eqs. (6.25), (6.26).
Fig. 16. Vertical isentropic sea-air profiles (almost vertical thin lines) of temperature for different SST values between 0 and 40°C at standard-ocean salinity and sea-level pressure of 101 325 Pa, to- gether with the related isentropic condensation levels (ICL) of liq- uid water and ice (bold lines). Below the freezing temperature of seawater, sea air is saturated at the sea surface. The slope of the isentropes is given by the adiabatic lapse rate, Eqs. (6.25), (6.26).
Table 1. Two selected radiosonde profiles taken off Morocco in September 1992 on r/v “A. v. Humboldt”. The sea-surface salinity SSS is given on the 1978 Practical Salinity Scale (PSS-78) which is converted to the Absolute Salinity S, =SSSxwups used in this paper by the factor ups =(35.16504/35) gkg—! (Millero et al., 2008; IOC, 2010).
Table 1. Two selected radiosonde profiles taken off Morocco in September 1992 on r/v “A. v. Humboldt”. The sea-surface salinity SSS is given on the 1978 Practical Salinity Scale (PSS-78) which is converted to the Absolute Salinity S, =SSSxwups used in this paper by the factor ups =(35.16504/35) gkg—! (Millero et al., 2008; IOC, 2010).
Fig. 17. Isentropic freezing level (IFL) and melting level (IML) of sea air with standard-ocean salinity (35.16504 g kg!) as a function of the sea-surface temperature (SST), computed from Eqs. (9.3), (9.4) with entropy and air fraction computed from Eqs. (6.4) and (6.18). At pressures higher than IFL, the condensed part is liquid, at pressures lower than IML, the condensate is ice. Between the two curves, the solid, liquid and gas phase of water coexist. Both curves begin at the condensation level (ICL), Fig. 16, and do not reach the sea surface. Note that the local temperature (Fig. 18) is always very near 0°C in the mixed-phase region between IML and JFL.
Fig. 17. Isentropic freezing level (IFL) and melting level (IML) of sea air with standard-ocean salinity (35.16504 g kg!) as a function of the sea-surface temperature (SST), computed from Eqs. (9.3), (9.4) with entropy and air fraction computed from Eqs. (6.4) and (6.18). At pressures higher than IFL, the condensed part is liquid, at pressures lower than IML, the condensate is ice. Between the two curves, the solid, liquid and gas phase of water coexist. Both curves begin at the condensation level (ICL), Fig. 16, and do not reach the sea surface. Note that the local temperature (Fig. 18) is always very near 0°C in the mixed-phase region between IML and JFL.
Fig. 18. Vertical isentropic sea-air profiles of temperature for differ- ent SST values between —40 and +40 °C at standard- ocean salinity 35.16504 gkg~!) and sea-level pressure (101 325 Pa), beyond the related isentropic condensation level shown in Fig. 16. Note that he pressure range in Fig. 16 is compressed into a very thin layer at the bottom of this plot. The isentropic transition between liquid and solid condensate, between IFL and IML in Fig. 17, has an ex- tremely small lapse rate and appears here as a pressure jump, i.e. as an almost isothermal layer in which the solid, liquid and vapour phase of water coexist, Fig. 11b.
Fig. 18. Vertical isentropic sea-air profiles of temperature for differ- ent SST values between —40 and +40 °C at standard- ocean salinity 35.16504 gkg~!) and sea-level pressure (101 325 Pa), beyond the related isentropic condensation level shown in Fig. 16. Note that he pressure range in Fig. 16 is compressed into a very thin layer at the bottom of this plot. The isentropic transition between liquid and solid condensate, between IFL and IML in Fig. 17, has an ex- tremely small lapse rate and appears here as a pressure jump, i.e. as an almost isothermal layer in which the solid, liquid and vapour phase of water coexist, Fig. 11b.
Fig. 19. Entropy of two radiosonde profiles taken on the subtropical Atlantic off Morocco, Table 1, computed from Eq. (5.16), neglect- ing any unknown condensed fraction, compared with the related sea-air entropy, computed from ocean surface properties by means of the equilibrium Eq. (6.4) and shown here as vertical lines. The large difference in the upper panel probably results from strong off- shore wind that also carried Sahara dust observed at this time.
Fig. 19. Entropy of two radiosonde profiles taken on the subtropical Atlantic off Morocco, Table 1, computed from Eq. (5.16), neglect- ing any unknown condensed fraction, compared with the related sea-air entropy, computed from ocean surface properties by means of the equilibrium Eq. (6.4) and shown here as vertical lines. The large difference in the upper panel probably results from strong off- shore wind that also carried Sahara dust observed at this time.
Fig. 20. Profiles of air fraction and entropy of the radiosonde pro- files AEROSE 04030715 taken on 7 March 2004 15:47 UTC in the tropical Atlantic at 9.85797° N, 33.11862° E showing an isen- tropic surface layer (Courtesy Nicholas R. Nalli, NOAA). The ver. tical lines indicate the values computed for the humid-air part from the sea-air equilibrium at the observed sea-surface temperature of 26.28°C and salinity of 36.22 g/kg, Eqs. (6.4) and (6.18). The air fraction A on the abscissa relates to specific humidity, g=1— A.
Fig. 20. Profiles of air fraction and entropy of the radiosonde pro- files AEROSE 04030715 taken on 7 March 2004 15:47 UTC in the tropical Atlantic at 9.85797° N, 33.11862° E showing an isen- tropic surface layer (Courtesy Nicholas R. Nalli, NOAA). The ver. tical lines indicate the values computed for the humid-air part from the sea-air equilibrium at the observed sea-surface temperature of 26.28°C and salinity of 36.22 g/kg, Eqs. (6.4) and (6.18). The air fraction A on the abscissa relates to specific humidity, g=1— A.
Table A1. Chemical composition of sea salt (Millero et al., 2008). Mole fractions are exact by definition, mass fractions are rounded to seven digits behind the period. Uncertainties of the molar masses (Wieser, 2006) are given in brackets.
Table A1. Chemical composition of sea salt (Millero et al., 2008). Mole fractions are exact by definition, mass fractions are rounded to seven digits behind the period. Uncertainties of the molar masses (Wieser, 2006) are given in brackets.
Table A2. Chemical composition of dry air. Mole fractions from Picard et al. (2008) except for Nz which was adjusted by subtracting all other mole fractions from 1 (Picard et al., 2008). Uncertainties of the molar masses (Wieser, 2006) are given in brackets.
Table A2. Chemical composition of dry air. Mole fractions from Picard et al. (2008) except for Nz which was adjusted by subtracting all other mole fractions from 1 (Picard et al., 2008). Uncertainties of the molar masses (Wieser, 2006) are given in brackets.
Table B1. Coefficients and exponents for the ideal-gas part (B2) for dry air (Lemmon et al., 2000). Here, we have adjusted the co- efficients ng and no to the reference state conditions of zero en- tropy and zero enthalpy of dry air at the standard ocean surface state, 0°C and 101325Pa. The originally published values are n{ = —13.841 928 076 and n¥ = 17.275 266 575. Appendix C Thermodynamic potential of fluid water The specific Helmholtz energy for fluid (gaseous and liquid) water is (IAPWS, 2009a; Wagner and Pru, 2002),
Table B1. Coefficients and exponents for the ideal-gas part (B2) for dry air (Lemmon et al., 2000). Here, we have adjusted the co- efficients ng and no to the reference state conditions of zero en- tropy and zero enthalpy of dry air at the standard ocean surface state, 0°C and 101325Pa. The originally published values are n{ = —13.841 928 076 and n¥ = 17.275 266 575. Appendix C Thermodynamic potential of fluid water The specific Helmholtz energy for fluid (gaseous and liquid) water is (IAPWS, 2009a; Wagner and Pru, 2002),
Table C1. Coefficients of the ideal part (C3). Note that the orig- inally published values (Wagner and Pru, 2002) of the adjustable coefficients nf and n5 are slightly different from those given here. Those given here are preferred (IAPWS, 2009b; Feistel et al., 2008b).
Table C1. Coefficients of the ideal part (C3). Note that the orig- inally published values (Wagner and Pru, 2002) of the adjustable coefficients nf and n5 are slightly different from those given here. Those given here are preferred (IAPWS, 2009b; Feistel et al., 2008b).
Table B2. Coefficients and exponents for the fundamental Eq. (B3) for dry air (Lemmon et al., 2000). The residual part of Eq. (C1) has the form
Table B2. Coefficients and exponents for the fundamental Eq. (B3) for dry air (Lemmon et al., 2000). The residual part of Eq. (C1) has the form
The temperature 7 is measured on the ITS-90 scale. The range of validity is 130-1273 K without the extension (C4), i.e. with 6%*(t) =0. The range can be extended to include the region 50-130 K with a correction function added in this temperature range, where 7g=130K , E=0.278296458178592, © =T7¢/Te. At t=e, @*(r) is zero, as well as its first, second, third and fourth temperature derivatives. This correction has been de- termined such that when applied to Cooper's formula used in IAPWS-95, it results in a fit to the heat capacity data of Woolley (1980) between 50 and 130K with an rm.s. devia- tion of 6x 10~* in c,/Rw. This extension formula has been developed particularly for implementation in the source-code ibrary and is not published elsewhere.
The temperature 7 is measured on the ITS-90 scale. The range of validity is 130-1273 K without the extension (C4), i.e. with 6%*(t) =0. The range can be extended to include the region 50-130 K with a correction function added in this temperature range, where 7g=130K , E=0.278296458178592, © =T7¢/Te. At t=e, @*(r) is zero, as well as its first, second, third and fourth temperature derivatives. This correction has been de- termined such that when applied to Cooper's formula used in IAPWS-95, it results in a fit to the heat capacity data of Woolley (1980) between 50 and 130K with an rm.s. devia- tion of 6x 10~* in c,/Rw. This extension formula has been developed particularly for implementation in the source-code ibrary and is not published elsewhere.
Table C2. Coefficients of the residual part (C5).
Table C2. Coefficients of the residual part (C5).
Table C4. Coefficients of the residual part (C5). gases, Eq. (H9). At higher densities, the corresponding Helmholtz energy of the real-gas mixture is
Table C4. Coefficients of the residual part (C5). gases, Eq. (H9). At higher densities, the corresponding Helmholtz energy of the real-gas mixture is
Table C3. Coefficients of the residual part (C5).
Table C3. Coefficients of the residual part (C5).
Here, (ars) is the shorthand notation for the Na nested volume integrations dr} dr>...dr,,, and similarly for ary), For the pure components, F“ and FV, the formu- ations of Lemmon et al. (2000) and IA PWS-95 are available from the literature. They provide empirical approximate ex- pressions for the integrals (D4), 04, and (D5), OY, valid up to very high densities of either dry air or pure water. For the mixed term, only the lowest-order cross-virial coefficients The configuration integrals 9“, OY and QA of the canon- ical partition function are defined in terms of the pair interaction potentials w between the particles of air at the positions r“ and those of vapour at r’, as (B =1/kT):
Here, (ars) is the shorthand notation for the Na nested volume integrations dr} dr>...dr,,, and similarly for ary), For the pure components, F“ and FV, the formu- ations of Lemmon et al. (2000) and IA PWS-95 are available from the literature. They provide empirical approximate ex- pressions for the integrals (D4), 04, and (D5), OY, valid up to very high densities of either dry air or pure water. For the mixed term, only the lowest-order cross-virial coefficients The configuration integrals 9“, OY and QA of the canon- ical partition function are defined in terms of the pair interaction potentials w between the particles of air at the positions r“ and those of vapour at r’, as (B =1/kT):
Table D1. Coefficients of the virial coefficients, Eqs. (D16-D18). are known. We introduce the binary air-vapour correlation function,
Table D1. Coefficients of the virial coefficients, Eqs. (D16-D18). are known. We introduce the binary air-vapour correlation function,
In the lowest order in density of this series, the exponentials of Eqs. (D4), (D5) and (D8) reduce to unity, and the remain- ing volume integrals of
In the lowest order in density of this series, the exponentials of Eqs. (D4), (D5) and (D8) reduce to unity, and the remain- ing volume integrals of
which defines the virial coefficients B and C as functions of the temperature. The 2nd air-air virial coefficient follows to be a volume integral over the binary air-air correlation func- tion (Reif, 1965; Kammer and Schwabe, 1971; Landau and Lifschitz, 1987), in the form
which defines the virial coefficients B and C as functions of the temperature. The 2nd air-air virial coefficient follows to be a volume integral over the binary air-air correlation func- tion (Reif, 1965; Kammer and Schwabe, 1971; Landau and Lifschitz, 1987), in the form
Fig. D1. Deviations of values for the saturation partial pressure of water calculated from different models to experimental data given by Pollitzer and Strebel (1924) at 323K and Webster (1950) at 273 and 288 K, Hyland and Wexler (1973) at 303, 313, and 323K, and Hyland (1975) at 343K, as well as Wylie and Fisher (1996) at 293K, 323K and 348 K, plotted over the total density. Considered models: “this work”, calculated from Eq. (7.5), “fmix=0” - this work without the cross-virial coefficients B and C, i.e. neglecting F™X=0, Eq. (D8). This result is consistent with earlier studies by Herrmann et al. (2009a, b, c, 2010); the virial model of Hyland and Wexler (1983a, b) is included in the latter studies.
Fig. D1. Deviations of values for the saturation partial pressure of water calculated from different models to experimental data given by Pollitzer and Strebel (1924) at 323K and Webster (1950) at 273 and 288 K, Hyland and Wexler (1973) at 303, 313, and 323K, and Hyland (1975) at 343K, as well as Wylie and Fisher (1996) at 293K, 323K and 348 K, plotted over the total density. Considered models: “this work”, calculated from Eq. (7.5), “fmix=0” - this work without the cross-virial coefficients B and C, i.e. neglecting F™X=0, Eq. (D8). This result is consistent with earlier studies by Herrmann et al. (2009a, b, c, 2010); the virial model of Hyland and Wexler (1983a, b) is included in the latter studies.
@ Normal salinity represents the salinity of KCl-normalized seawater (Millero et al., 2008). Table E1. Special constants and values used in Eq. (E2).
@ Normal salinity represents the salinity of KCl-normalized seawater (Millero et al., 2008). Table E1. Special constants and values used in Eq. (E2).
Table E2. Coefficients of the saline part of the Gibbs function (E2). Coefficients not contained in this table have the value g; ;, =0.
Table E2. Coefficients of the saline part of the Gibbs function (E2). Coefficients not contained in this table have the value g; ;, =0.
Table F1. Coefficients of the Gibbs potential (F1). Table G1. The partial derivatives of the Helmholtz function, f(A,7,), expressed in terms of thermodynamic coefficients. «7: isotherma compressibility, a: thermal expansion coefficient, cy: specific isochoric heat capacity.
Table F1. Coefficients of the Gibbs potential (F1). Table G1. The partial derivatives of the Helmholtz function, f(A,7,), expressed in terms of thermodynamic coefficients. «7: isotherma compressibility, a: thermal expansion coefficient, cy: specific isochoric heat capacity.
Table G2. Partial derivatives of the Gibbs function, g, expressed as partial derivatives of the Helmholtz function, f. Subscripts indicat. partial derivatives with respect to the respective variables.
Table G2. Partial derivatives of the Gibbs function, g, expressed as partial derivatives of the Helmholtz function, f. Subscripts indicat. partial derivatives with respect to the respective variables.
@ The quantity ¢47 appearing here is related to the thermodiffusion coefficient (Landau and Lifschitz, 1974, §58) but has no common name or symbol. Table G3. Partial derivatives of the enthalpy potential function, h, expressed as partial derivatives of the Gibbs function, g. Subscript: indicate partial derivatives with respect to the respective variables.I’: adiabatic lapse rate, «;: isentropic compressibility. The ideal-gas Helmholtz function of water vapour has the similar form,
@ The quantity ¢47 appearing here is related to the thermodiffusion coefficient (Landau and Lifschitz, 1974, §58) but has no common name or symbol. Table G3. Partial derivatives of the enthalpy potential function, h, expressed as partial derivatives of the Gibbs function, g. Subscript: indicate partial derivatives with respect to the respective variables.I’: adiabatic lapse rate, «;: isentropic compressibility. The ideal-gas Helmholtz function of water vapour has the similar form,
_ The ideal-gas part of the Helmholtz function for dry air, Eq. (B2), takes the form Having computed the pressure from the density derivative of Eq. (H3), as For the Helmholtz function of humid air we obtain from Eq. (5.2) in this limit, Eq. (D1), sufficiently accurate for certain ap of these relations are well known repeat the formulas here for easy re tities in the more general case of tl ular for their adjustable constants the real-gas formulation, Table H1. dispensable for applications such a plication purposes. Most (Guggenheim, 1967); we ference to the same quan- he real gas, and in partic- chosen consistently with This consistency is in- s the fugacity (10.8). The Helmholtz functions (H1) and (H2 are equivalent to the ideal parts (B2) and (C2) except that the temperature dependencies of the heat capacities are not specified here and may be re- placed by simpler expressions if this is appropriate.
_ The ideal-gas part of the Helmholtz function for dry air, Eq. (B2), takes the form Having computed the pressure from the density derivative of Eq. (H3), as For the Helmholtz function of humid air we obtain from Eq. (5.2) in this limit, Eq. (D1), sufficiently accurate for certain ap of these relations are well known repeat the formulas here for easy re tities in the more general case of tl ular for their adjustable constants the real-gas formulation, Table H1. dispensable for applications such a plication purposes. Most (Guggenheim, 1967); we ference to the same quan- he real gas, and in partic- chosen consistently with This consistency is in- s the fugacity (10.8). The Helmholtz functions (H1) and (H2 are equivalent to the ideal parts (B2) and (C2) except that the temperature dependencies of the heat capacities are not specified here and may be re- placed by simpler expressions if this is appropriate.
Table H1. Consistently chosen adjustable constants and related A et of the ideal-gas formulas with respect to the ideal-gas parts of the formulations for dry air, ADIPE", {Lemmon et al., 2000), Eq. (B2), oe water vapour, y AEWS95", Eq. (C3), and their reference states. The densities of and oy satisfy the ideal-gas tions oh Rk T) = Po and py h Beet = Pt. For compari- son, if the latest values M4 =28.96546gmol—!, mW =18.015268gmol—! and R=8.314472J mol-! K—! are used rather than the older values, the related ideal-gas densities at the reference points change to p4-!4 = Mg Po/(RT 9)=1.2922935073108kgm~3 and pYid = Mw Pi/(RT )=0.00485171538112533 kg m2, we derive the ideal-gas Gibbs function of humid air, gud — fav id pAV.id (AV Bg, (5.12), as
Table H1. Consistently chosen adjustable constants and related A et of the ideal-gas formulas with respect to the ideal-gas parts of the formulations for dry air, ADIPE", {Lemmon et al., 2000), Eq. (B2), oe water vapour, y AEWS95", Eq. (C3), and their reference states. The densities of and oy satisfy the ideal-gas tions oh Rk T) = Po and py h Beet = Pt. For compari- son, if the latest values M4 =28.96546gmol—!, mW =18.015268gmol—! and R=8.314472J mol-! K—! are used rather than the older values, the related ideal-gas densities at the reference points change to p4-!4 = Mg Po/(RT 9)=1.2922935073108kgm~3 and pYid = Mw Pi/(RT )=0.00485171538112533 kg m2, we derive the ideal-gas Gibbs function of humid air, gud — fav id pAV.id (AV Bg, (5.12), as
Here, the pressure as an independent variable is written with- out superscripts, P = PAY -i4, The related entropy follows as
Here, the pressure as an independent variable is written with- out superscripts, P = PAY -i4, The related entropy follows as
Table 11. Numerical check values for the Helmholtz function fAY and its derivatives, Eq. (D14), as well as selected properties, Eqs. (5 5.24), for saturated air, A = A%*, with respect to ice at 200K, and with respect to liquid water at 300 and 400 K. The humid-air densit p= ph¥ = pA oY =mAV v= (m4 rm) /V, Appendix J
Table 11. Numerical check values for the Helmholtz function fAY and its derivatives, Eq. (D14), as well as selected properties, Eqs. (5 5.24), for saturated air, A = A%*, with respect to ice at 200K, and with respect to liquid water at 300 and 400 K. The humid-air densit p= ph¥ = pA oY =mAV v= (m4 rm) /V, Appendix J
Table I3. Numerical check values for the cross-virial part, f mix and the cross-virial coefficients, Baw, Caaw, Caww, Of the Helmholtz function f4V and its derivatives, Eq. (D14), for saturated air, A=AS*, with respect to ice at 200K, and with respect to liquid water at 300 and 400K, corresponding to Table I1. The numerical functions evaluated here at given points (A, 7, p“V) are defined in Eqs. (D15-D18). The humid-air density is p = pV =pA+pV = mAV / V= (mA +m) /V. TableI2. Numerical check values for the dry-air part, f*, and the water-vapour part, f”, of the Helmholtz function f4V and its derivatives, Eq. (D14), for saturated air, A = AS*, with respect to ice at 200K, and with respect to liquid water at 300 and 400K, corresponding to Table 11. The numerical functions evaluated here at given points (A, T, pV) are defined in Eqs. (B1) and (C1). The humid-air density is AV A Vv AV... Lf RAR. VN Oe
Table I3. Numerical check values for the cross-virial part, f mix and the cross-virial coefficients, Baw, Caaw, Caww, Of the Helmholtz function f4V and its derivatives, Eq. (D14), for saturated air, A=AS*, with respect to ice at 200K, and with respect to liquid water at 300 and 400K, corresponding to Table I1. The numerical functions evaluated here at given points (A, 7, p“V) are defined in Eqs. (D15-D18). The humid-air density is p = pV =pA+pV = mAV / V= (mA +m) /V. TableI2. Numerical check values for the dry-air part, f*, and the water-vapour part, f”, of the Helmholtz function f4V and its derivatives, Eq. (D14), for saturated air, A = AS*, with respect to ice at 200K, and with respect to liquid water at 300 and 400K, corresponding to Table 11. The numerical functions evaluated here at given points (A, T, pV) are defined in Eqs. (B1) and (C1). The humid-air density is AV A Vv AV... Lf RAR. VN Oe
TableJ 1. Formula symbols used in the paper.
TableJ 1. Formula symbols used in the paper.
Table J 2. Abbreviations used in the paper.
Table J 2. Abbreviations used in the paper.
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